I've 2 physical machines and both are running CentOS 6.10 x86_64. Both seem to be running just fine with the exception of Firefox. And from basic analysis if seems to come down to how things were installed:

Machine A:
Installed using full CentOS 6.7 x86_64 DVD and during the install Customize Now was used streamlining what was installed. So the Gnome GUI was available after install including Firefox. The inittab was altered from using '5' to '3' so when I want the GUI I just enter 'startx'. Then 'yum -y update was used to bring it up to CentOS 6.10 and rebooted. All is well including Firefox.

Machine B:
Installed using the CentOS 6.10 Minimal DVD using the Server World tutorial. I needed to have the Gnome GUI on this machine so I followed their instructions to use groupinstall that included 'Internet Applications' and 'Internet Browser' groups. Performed yum -y update to bring the system up to the latest. Rebooted and started the GUI. Went to applications to start Firefox got the Firefox starting message but that was all.

OK, right after I posted my prior reply I entered the same command (/usr/bin/firefox) from machine A and it worked just fine. A difference I saw between to two machines is the Java being used (the Java hasn't been upgraded to jdk1.8.0_202):